We study the joint distribution of stochastic events described by (X,Y,N), where N has a 1-inflated (or deflated) geometric distribution and X, Y are the sum and the maximum of N exponential random variables. Models with similar structure have been used in several areas of applications, including actuarial science, finance, and weather and climate, where such events naturally arise. We provide basic properties of this class of multivariate distributions of mixed type, and discuss their applications. Our results include marginal and conditional distributions, joint integral transforms, moments and related parameters, stochastic representations, estimation and testing. An example from finance illustrates the modeling potential of this new model.

中文翻译：

---

随机事件的新三变量模型

我们研究由（X，Y，N）描述的随机事件的联合分布，其中N具有1膨胀（或缩小）的几何分布，而X，Y是N个指数随机变量的总和和最大值。具有类似结构的模型已用于多个应用领域，包括精算科学，金融，天气和气候等自然发生的领域。我们提供此类混合类型的多元分布的基本属性，并讨论其应用。我们的结果包括边际和条件分布，联合积分变换，弯矩和相关参数，随机表示，估计和测试。财务中的一个例子说明了这种新模型的建模潜力。